4: Uncertainty and Probability

“The scientist has a lot of experience with ignorance and doubt and uncertainty, and this experience is of very great importance, I think.

When a scientist does not know the answer to a problem, he is ignorant. When he has a hunch as to what the result is, he is uncertain. And when he is pretty damn sure of what the result is going to be, he is still in some doubt.

We have found it of paramount importance that in order to progress, we must recognize our ignorance and leave room for doubt.”

Richard Feynman, 1988

CQ: If you are caught riding the bus without a ticket, you will be fined 40 Euros. What is the expected cost of getting caught? If a ticket costs 2 Euros, does it make sense to buy a ticket? Do you buy a ticket? Why or why not?

CQ:

Flip a coin twice and record the outcome – did you get at least one head?

Combine all of class’ tosses to see what fraction came up heads

BBC 6 June 2013

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Gaussian distribution

PAN 4.1: Probability of observing specific values from a Gaussian parent distribution.

Men

Women

Distribution of heights: men vs women

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EXCEL: normal distribution

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http://www.geokem.com/earths_average_composition.html

Basalt generally has a composition of 45-55 wt% SiO2, 2-6 wt% total alkalis, 0.5-2.0 wt% TiO2, 5-14 wt% FeO and 14 wt% or more Al2O3. Contents of CaO are commonly near 10 wt%, those of MgO commonly in the range 5 to 12 wt%.

Oceanic crust variability

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Sum of two-point averages

Sum of 3-point averages Sum of 16-point averageshttp://www.statisticalengineering.com/central_limit_theorem.htm

Starting nonnormal distribution

Numerical demonstration of Central Limit Theorem

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PAN 4.2: Histogram of the results of drawing N samples from a Gaussian parent distribution with mean zero and a unit standard deviation.

For small numbers of samples the observed distribution can look quite different from the parent distribution and the sample mean ' differs from that of the parent distribution.

As the number of samples increases, the observed distribution looks increasingly like the parent distribution. (Stein and Wysession, 2003)

CQ: A challenge for medical researchers involves "cancer clusters," communities that

have higher-than-expected cancer rates. Some clusters would be expected to occur purely

from chance, because of 100 communities, 5% should have rates that are significantly higher at 95% confidence, etc. How could one try to

identify which clusters result from factors other than chance?

PAN 4.3: Maximum rate of motion in and across the New Madrid seismic zone shown by successively more precise GPS measurements. (Calais and Stein, 2009)

Shallow uncertainty - we don’t know what will happen, but know the odds (pdf). The past is a good predictor of the future. We can make math models that work well.

Deep uncertainty - we don’t know the odds (pdf). The past is a poor predictor of the future. We can make math models, but

they won’t work well.

Shallow uncertainty is like estimating the chance that a batter will get a hit. His batting average is a good predictor.

Deep uncertainty is like trying to predict the winner of the World Series next baseball season. Teams' past performance give only limited insight into the future.

CQ: In 2003, a magnitude 6.6 earthquake destroyed much of the city of Bam, Iran. Much of the city, including its famous 2000-year-old citadel, was made of mud brick that is very vulnerable to earthquake shaking and thus was seriously damaged. What insights does the city's history give for how often on average such earthquakes happen? What can you say and not say?

M 6.6 27,000 deaths

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